Principled BSDF (and other shaders) do not render textures with the Eevee engine
OK I eventually fixed it.
I ran debug mode and got the exact same error as on this thread posted this morning:
https://developer.blender.org/T83272
This is what they said:

Next time, please follow the submission template and attach the debug
log and system info as file.
Please try to install a previous version of the graphics driver prior
to Adrenalin 2020 Edition 20.11.2 Optional, 27.20.14501.18003 and
report back if this resolves the issue.

I downgraded my graphics to 20.11.1 and it worked fine.